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Exam Measure Theoretic Probability Korteweg-de Vries Instituut voor wiskunde, UvA Lecturer: Sonja Cox Date/time: Wednesday, January 7th 2015, 14:00-17:00 Student name: Student number: Student affiliation (university): Exercise Max points Points scored (1st correction) Points scored (2nd correction) 1 4 2 12 3 14 4 16 5 8 total 54 NOTE: Unless indicated otherwise, all results (theorems/propositions/lemmas) in the MTP lecture notes may be used to answer the exam questions. Please indicate what results you are using. Exercise 1 (4pt) Let (Xn )∞ variables n=1 be a sequence of independent, identically distributed random Q on a probability space (Ω, F, P). Assume EX1 = 0. Prove: limn→∞ nk=1 eXk /n = 1 a.s. Exercise 2 (12pt) For A ⊂ B([−1, 1]) define −A = {x ∈ R : − x ∈ A}. Let λ denote the Lebesgue measure on B([−1, 1]). Define A = {A ∈ B([−1, 1]) : λ(A) = λ(−A)}. i. (3pt) Prove that A is a d-system. ii. (1 pt) Prove that A = B([−1, 1]). 1 iii. (4pt) Prove: if f :R[−1, 1] → R is a simple function and f (x) = −f (−x) for all 1 x ∈ [−1, 1], then −1 f (x) dx = 0. iv. (4pt) Prove: if f : [−1, R 1 1] → R is Lebesgue integrable and f (x) = −f (−x) for all x ∈ [−1, 1], then −1 f (x) dx = 0. Exercise 3 (14pt) Let (Xn )∞ n=1 be a sequence of independent, identically distributed random variables on a probability space (Ω, F, P) satisfying EX1 = 0 and EX12 = 1. Define F0 = {∅, Ω} and for n ∈ N define Fn = σ(X1 , . . . , Xn ) and Mn = n X 1 (X1 k · . . . · Xk ) k=1 (i.e., M1 = X1 , M2 = X1 + 21 X1 X2 , M3 = X1 + 12 X1 X2 + 13 X1 X2 X3 ). i. (4pt) Calculate EMn2 , n ∈ N. ∞ ii. (4pt) Prove that (Mn )∞ n=1 is an (Fn )n=1 -martingale. iii. (4pt) Provide hM i. iv. (2pt) Is (Mn )∞ n=1 uniformly integrable? Exercise 4 (18pt) Let (Ω, F, P) be a probability space. Let An,j ∈ F, n ∈ N0 , j ∈ {1, 2, 3, . . . , 2n }, be Sn such that for all n ∈ N0 it holds that 2j=1 An,j = Ω and ∀i, j ∈ {1, 2, 3, . . . , 2n }, i 6= j : An,i ∩ An,j = ∅ and An,i = An+1,2i−1 ∪ An+1,2i . (1) For n ∈ N0 define Fn = σ({An,j : j ∈ {1, 2, 3, . . . , 2n }). i. (2pt) Prove that (Fn )∞ n=0 is a filtration. ii. (3pt) Let µ : F → [0, ∞) be a probability measure on (Ω, F). Assume: If P(A) = 0 for some A ∈ F, then µ(A) = 0. (2) For n ∈ N0 define ( µ(A Mn (ω) = n,j ) , P(An,j ) ω ∈ An,j , P(An,j ) 6= 0; 0, otherwise. Prove that for all n ∈ N0 and all j ∈ {1, 2, 3, . . . , 2n } it holds that Z Z Mn+1 dP = Mn dP. An,j An,j 2 (3) (4) ∞ iii. (1pt) Explain why it follows that (Mn )∞ n=1 is (Fn )n=0 -martingale. R iv. (2pt) Prove that for all A ∈ Fn it holds that µ(A) = A Mn dP. v. (4pt) Assumption (2) is equivalent to: ∀ε > 0 ∃δ > 0 : if P(A) < δ for some A ∈ F, then µ(A) < ε. (5) Prove that (Mn )∞ n=1 is uniformly integrable. S vi. (4pt) Define F∞ =R σ( n∈N Fn ). Prove that there exists a M∞ ∈ L1 (Ω, F∞ , P) such that µ(A) = A M∞ dP for all A ∈ F∞ . Exercise 5 (10pt) Let (Xn )∞ n=1 be a sequence of independent, identically distributed random variables 1 on a probability space (Ω, F, P) satisfying P(X1 = 1) = P(X Pn 1 = −1) = 2 . For n ∈ N 1 let φn : R → C denote the characteristic function of √n k=1 Xk . In this exercise you may not use the Central Limit Theorem. You may use that by l’Hopital’s rule we have that lim x↓0 log(cos(ux)) −u sin(ux) −u2 cos(ux) u2 = lim = lim = − . x↓0 2x cos(ux) x↓0 2(cos(ux) − x sin(ux)) x2 2 i. (4pt) Prove that φn (u) = (cos(n−1/2 u))n for u ∈ R. ii. (4pt) Let γ : Ω → R be a standard Gaussian random variable and let φγ : R → 2 R denotePits characteristic function, i.e., φγ (u) = e−u /2 for all u ∈ R. Prove that √1n nk=1 Xk → γ weakly as n → ∞. Exercise Let (Xn )∞ n=1 be a sequence of independent, identically distributed random variables on a probability space (Ω, F, P) satisfying P(X1 = 1) =PP(X1 = −1) = 21 . For n ∈ N and k ∈ {1, . . . , n} let ak,n ∈ [0, ∞) be such that nj=1 a2n,j = 1 and define ξn,k = an,k Xk . i. Formulate the Lindeberg condition for (ξn,k )n∈N,k∈{1,...,n} . P ii. Provide explicit values of an,k , n ∈ N, k ∈ {1, . . . , n}, such that nk=1 ξn,k → γ weakly as n → ∞, where γ is a standard normal Gaussian random variable. Exercise Let (S, S) be a measurable space and let µ, ν : S → [0, ∞) be finite measures on (S, S). Assume that for all A ∈ S it holds that µ(A) = 0 implies ν(A) = 0. Prove that for all ε > 0 there exists a δ > 0 such that if µ(A) < δ, then ν(A) < ε . 3 Exercise Let (Ω, F, P) be a probability space and let An ∈ F, n ∈ N. Assume F = σ({An : n ∈ N}). For n ∈ N define Fn = σ({Ak : k ∈ {1, . . . , n}}). Let B ∈ F. i. Prove that limn→∞ E[1B |Fn ] = 1B a.s. ii. Does it hold that limn→∞ E|E[1B |Fn ] − 1B | = 0? Motivate your answer. Exercise Let (Xn )∞ n=1 be a sequence of independent, identically distributed random variables on a probability space (Ω, F, P) satisfying E(X12 ) < ∞. Let Yn → Y weakly as n → ∞. Prove that for all a, b ∈ R, a < b, it holds that lim inf n→∞ P(Yn ∈ (a, b)) > P(Y ∈ (a, b). 4